Along a 1000 meter long beach, you and two more ice cream vendors are setting up shop. You arrive first. Where do you place your cart?
Ice Cream Vendor etiquette dictates that no two vendors set up their carts closer than 10 meters from one another. Moreover, once you set up your shop, you must not move it for the rest of the day. Every part of the beach has the same density of customers, but they are lazy and will frequent only the ice cream stand closest to them.
Assuming that the other two vendors will arrive one after the other and that they will choose a location that will maximize their income, where should you set up your cart?
Have fun posting your solution! We will add ours in a couple of days.
Assume there was only one additional vendor coming, where would you set up shop then?
Somewhat surprisingly, if there is only one other vendor, we would set up at 500 meters and they at 490 (or 510), giving us 505 meters of business and taking 495 meters themselves.
For two more vendors, we best set up at 252 meters (or 748). The second vendor then rationally sets up at 748 meters, making it unattractive for the third vendor to set up shop in [758,1000] (and also [0,242]). The third vendor will want to set up shop somewhere in the middle in [262, 738], as 248 meters of business are better than the 247 they could maximally get otherwise.
Worst for us is if they go to 262, which would leave us with 25.7% of the business, and vendor 2 with 49.5%. No matter where in the middle they set up shop, vendor 3 walks away with 24.8% of the business.